We propose a simplicial complex convolutional neural network (SCCNN) to learn data representations on simplicial complexes. It performs convolutions based on the multi-hop simplicial adjacencies via common faces and cofaces independently and captures the inter-simplicial couplings, generalizing state-of-the-art. Upon studying symmetries of the simplicial domain and the data space, it is shown to be permutation and orientation equivariant, thus, incorporating such inductive biases. Based on the Hodge theory, we perform a spectral analysis to understand how SCCNNs regulate data in different frequencies, showing that the convolutions via faces and cofaces operate in two orthogonal data spaces. Lastly, we study the stability of SCCNNs to domain deformations and examine the effects of various factors. Empirical results show the benefits of higher-order convolutions and inter-simplicial couplings in simplex prediction and trajectory prediction.
翻译:我们建议一个简单复杂的神经神经网络(SCCNN)来学习关于简易综合体的数据表达方式,通过共同面孔和共同面孔独立地进行基于多希望的简易对称的演进,并捕捉简化组合,将技术的状态加以概括化,在研究简化域和数据空间的对称性时,它被显示为相互调和方向等同性,从而结合了这种感性偏见。根据霍奇理论,我们进行光谱分析,以了解SCCNN如何管理不同频率的数据,表明通过面部和共同面部的演进在两个或多个数据空间运作。最后,我们研究SCCNN的稳定性,以控制变形,并研究各种因素的影响。经验性结果显示,在简单预测和轨迹预测中,较高顺序的共变异和简单组合的好处。